On double-pulse stability criteria with damping

A simple necessary condition for the stability of a linear dynamic system with elastic and damping characteristics which vary periodically with the same period is derived. General explicit necessary and sufficient conditions for stability are then developed for a double-pulse system. Such a system can be characterized by a pair of eigenvalues, or complex frequencies, corresponding to each half-period, and the stability of this system depends only on these complex frequencies. It is shown that a necessary, though not sufficient, condition for the stability of any such system is that the arithmetic mean of the real parts of all four of the complex frequencies over an entire period be negative or zero. This is shown to be true, more generally, for an $\mathcal {n}$-pulse system. The physical significance of the results is discussed, and numerical examples are given.